Visualizing Variation in Numerical Data: Dotplots, Histograms, and Density Plots

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Section 2.1 Visualizing Variation in Numerical Data

Exploring a Distribution of Numerical Data

Understanding how numerical data varies is fundamental in statistics. The distribution of a sample organizes the observed values and their frequencies, helping us identify patterns and summarize the data effectively. Visual tools such as dotplots, histograms, and density plots are essential for this purpose.

What Is a Distribution?

Distribution: A list that records the values observed in a sample and the frequency (count) of each value.
Frequency: The number of times a value occurs in the data set.
Distributions are central to data analysis, as they organize and summarize variation within the data.

Components of a Numerical Distribution

Shape: The visual appearance of the distribution (e.g., symmetric, skewed).
Center: The typical value around which data clusters (e.g., mean, median).
Variability: The spread or dispersion of the data values.

Examining a Distribution

To analyze a distribution, follow a two-step process:

Visualize the data: Use graphs to display the data and reveal patterns.
Summarize the data: Describe the shape, center, and spread, and identify any unusual values (outliers).

Visualizing Data: Dotplots

Dotplots are a simple and effective way to display individual data values and their frequencies. Each dot represents one observation, stacked above the corresponding value on a number line.

Construction steps:
1. Draw a horizontal axis for possible values.
2. Place a dot above each value for every observation.
3. Label the axis with the variable name.
Advantages: Shows individual values, helps investigate shape.
Disadvantages: Not ideal for large data sets with many unique values.

Example: Number of goals scored by San Diego Wave Football Club players in the 2022 season.

Dotplot of goals scored by San Diego Wave Football Club players

Dotplot Example: Book Costs

Dotplots can be used to analyze the cost of textbooks at a college bookstore, helping to identify unusually expensive or inexpensive books.

Dotplot of textbook costs at a college bookstore

4 textbooks cost $150 or more.
8 out of 23 textbooks (34.8%) cost $50 or less.
The textbook costing close to $300 is an outlier.

Visualizing Data: Histograms

Histograms group data into intervals called bins and display the frequency or relative frequency of values within each bin. They are useful for large data sets and for visualizing the general shape of the distribution.

Frequency Histogram: Vertical axis shows frequency; horizontal axis is numerical.
Relative Frequency Histogram: Vertical axis shows relative frequency (proportion or percent).
Bin Width: The choice of bin width affects the level of detail shown. Too narrow shows too much detail; too wide hides important features.

Example: Number of days to maturity for short-term investments.

Frequency and relative frequency histograms for short-term investments

Histogram Example: Goals Scored

Histograms can be constructed by creating bins (e.g., 0–2, 2–4, 4–6, etc.) and counting the number of observations in each bin. The shape of the histogram changes with bin width.

Histogram of goals scored by San Diego Waves FC players Histogram with narrow bin width Histogram with wide bin width

Changing bin width can reveal or obscure patterns in the data.

Advantages and Disadvantages of Histograms

Advantages: Good for large data sets, helps focus on general shape, easy to spot outliers.
Disadvantages: Individual data values are not visible, shape affected by bin width.

Visualizing Data: Density Plots

Density plots are high-tech graphical summaries that provide a smooth picture of the distribution. They are generated using computational algorithms and are similar to smoothed histograms. The vertical axis represents density, which is analogous to relative frequency.

Advantages: Smooths out meaningless fluctuations, allows comparison of multiple distributions.
Disadvantages: Requires technology, not possible by hand.

Example: Density plot alongside relative frequency histogram for goals scored.

Density plot of goals scored by San Diego Waves FC players Relative frequency histogram of goals scored by San Diego Waves FC players

The density plot outlines the general shape of the histogram, indicating a large peak at 0 goals and a steep decline.

Summary Table: Comparison of Dotplots, Histograms, and Density Plots

Graph Type	Shows Individual Values?	Best For	Key Features
Dotplot	Yes	Small data sets	Displays each observation, reveals shape
Histogram	No	Large data sets	Groups data into bins, shows frequency/relative frequency
Density Plot	No	Comparing distributions, smoothing	Smoothed version of histogram, shows density

Key Formulas

Relative Frequency:
Percent:

Using Technology: TI-84 Calculator for Histograms

Enter data in L1.
Turn on Stat Plot and select histogram type.
Set Xlist to L1 and view the histogram.
Use TRACE to navigate and view information.

Conclusion

Visualizing variation in numerical data is crucial for understanding distributions. Dotplots, histograms, and density plots each offer unique advantages and are chosen based on the size and nature of the data set. Experimenting with bin widths and using technology can enhance the analysis and reveal important patterns.